An adaptive Newton-method based on a dynamical systems approach

Amrein, Mario; Wihler, Thomas (2014). An adaptive Newton-method based on a dynamical systems approach. Communications in Nonlinear Science and Numerical Simulation, 19(9), pp. 2958-2973. Elsevier 10.1016/j.cnsns.2014.02.010

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The traditional Newton method for solving nonlinear operator equations in Banach spaces
is discussed within the context of the continuous Newton method. This setting makes it
possible to interpret the Newton method as a discrete dynamical system and thereby to
cast it in the framework of an adaptive step size control procedure. In so doing, our goal
is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

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